Teaching Entropy Is Simple.pdf

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Foreword

The following is an expanded version of a talk written for AP teachers andequally useful for those teaching first-year general chemistry, of course. Myformat in a lecture ms. is to underline unduly and to type words in italics or

boldface that I want to emphasize especially. These are bad form in print, butthey highlight important points in this introduction and so I have left a number ofthem.

Introduction

I admit that the title may be unsettling. Entropy isn't at allsimple inadvanced work. Even quantitative problems in a beginning course can bedifficult for students as well as for us to guide students through them. However,I think the basic qualitative ideas of the nature of energy and entropy aresurprisingly simple and sharing with students what we'll talk about here willactually answer the old question, "What is entropy, really?" That can changetheir whole attitude toward class work in the oft-dreaded topic ofthermodynamics.

Discarding the archaic idea of "disorder" in regard to entropy is essential.It just doesn't make scientific sense in the 21st century and its apparentconvenience often is flat-out misleading. As of November 2005, fifteen first-yearcollege texts have deleted entropy is disorder although a few still retainreferences to energy becoming disorderly. (This latter description is

meaningless, as I shall mention here, and discuss in detail in "Disorder ACracked Crutch For Supporting Entropy".) Most high school texts are written byunknown people working for publishers rather than by the individuals listed ontheir covers. Thus, they are slow to include changes in scientific concepts andyour HS text may still contain the obsolete entropy is disorder.

Steps to a simple fundamental view of entropy

First , we'll see how all spontaneous events from physical processes like

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dropping a ball to chemical reactions like the explosion of hydrogen in oxygen all all are basically due to energy.spreading.out. in the processes.Dispersing. Superficially, spreading out in some cases because the systemincreases in volume, or in other cases because the system is heated.,Fundamentally in all cases, dispersing energy in the sense that the moleculesenergies are in more arrangements ore different arrangements on a multitude ofquantized energy levels aftera process or reaction thatbefore.

Energy of all types spontaneously disperses if it is not hindered fromdoing so.

Second , entropy is the measure of energy dispersal, as a function oftemperature. In chemistry, the kind of energy that entropy measures ismotional energy of molecules that are translating (moving and colliding),rotating, and vibrating (atoms in a molecule moving as though the bonds weresprings) and phase change energy (enthalpy of fusion or vaporization). When

students realize that entropy is just a measuring device, a yardstick in a sense,then understanding it is no longer a vague 'big deal' although it is still anextremely important deal !

What entropy measures is how much energy is spread out in a process/T ORhow spread out the initial energy of a system becomes in that system (atconstant temperature). Exactly how entropy measures how much energy isdispersed in phase change is mathematically simple ( Hfusion or vaporization), in

standard state entropy (S0), in temperature change, (Cp dT/T) and in many

other applications, even though it is not awlays necessary to show your students

the details of calculation. Whenever entropy is encountered, energy becomingdispersed (involving T) is the function on which we should focus. However, thatviewpoint was silently buried in the simple equations that we were taught andto which in the past we have introduced our students. If we " follow the energyflow " then we and they can readily understand why and how entropy changes inall elementary thermodynamics.

In advanced work, in the many differential equations involving dS, therelation of energy dispersal to entropy change can be so complex as to be totallyobscured. But not in first-year thermodynamics!

Third, we'll look at examples of what "entropy" means in

how much energy is dispersed cases:the standard state "entropy" of any substance at 298 K from Tables(a very approximate index guide to the amount of energy that hasbeen spread out in a substance to heat it from 0 K to 298 K so thatit can exist energetically at 298 K.

(By putting entropy in quotes, I mean to imply that most

1.A.


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often "entropy" actually involves an "entropy c h a n g e ", themeasurement of a change in the amount of energy dispersalin a system (or surroundings) before a process and after aprocess. Only for perfect crystals at absolute zero, doesentropy mean a single measurement, i.e. 0 joules/K!)

the entropy change in a solid as it melts to a liquid or a liquid boils

or the converse, i.e. phase change;

2.

the entropy change in any system even as simple as the iron metalin frying pan as it is heated (and as it cools); and

3.

how spread out in a system energy becomes cases:the entropy change of a gas expanding into a vacuum; and4.the entropy change when ideal gases or ideal liquids mix, and aperfect solute (no enthalpy of dissolving) dissolves in a solvent.Both (4) and (5) involve no change in the amount of total motionalenergy (plus phase change energy) before the process and after,but their initial energy spontaneously becomes more spread out inthe larger (or somewhat different) volume in a process that makes

it available.

5.

B.

Fourth, after talking about examples of entropy change in terms ofmacro thermodynamics, i.e., qrev/T, we'll also look at what energy "spreading

out" or dispersing means in terms of molecular behavior, how the Boltzmannentropy equation quantitatively links energy dispersal to the number ofmicrostates in a system.

Finally, we'll see that Gibbs' G is more closely related to entropy than itis to energy. If G = H - TS is divided by T, the result is G/T, an entropyfunction. Thus, a better description than "free energy" for Gibbs' G is "the total

dispersible energy in the universe due to a chemical reaction" in parallel torealizing that the dqrevin dqrev/T that is "energy that has been or could be

dispersed" in a process.

Energy spontaneously disperses if it is not hindered from doing so

A ball lifted up above the floor has been given potential energy (PE),ordinarily by human action. When not hindered by one's grip, the ball will fall;its potential energy changes to kinetic energy that spreads out to pushing aside

the air, to a bit of sound, to a small amount of heat in the ball and the floor as itbounces and comes to rest. The ball's original localized potential energy hasbecome widely dispersed in different varieties of molecular motional energy.

A hot metal pan spontaneously disperses its energy to the air in a coolerroom. (This seemingly trivial example we shall see as a prototype of whathappens in any such spontaneous energy transfer from hotter to cooler a netincreasein the entropy of the combination of system and surroundings.)

If we dropped a bottle of nitroglycerin on a concrete floor, the mechanical


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shock could be greater than the activation energy that otherwise hinders thespontaneous decomposition of nitroglycerin. The "nitro" would then change intoother substances explosively, because some of its bond energy/enthalpy thatquantity not transferred to bonds in its reaction products is spread outextremely rapidly and thus enormously increases the vigorous motions of thegaseous product molecules. In summary: some of the potential energy (bondenthalpy) that was in the nitroglycerin becomes widely dispersed in themolecular movement of the products.

An ideal gas will spontaneously flow into an evacuated chamber andspread its original energy over the larger final volume; the speed of themolecules is unchanged but their energy becomes dispersed more widely throughout a larger domain.

Hydrogen and oxygen in a closed chamber will remain unchanged foryears and probably for millennia. Despite their larger combined bond enthalpies(higher potential energy) compared to water's (i.e., the energetic orthermodynamic reason why they should react to yield water), their spontaneous

reaction to form water is hindered by an activation energy. However, if a sparkis introduced, they will react explosively to dissipate some of their combinedbond energy in causing the products of the reaction water molecules tomove extremely rapidly as motional energy that we sense as high-temperaturesteam. Thereby energy is further spread out to the surroundings.

Those illustrations lead to a profound generalization: In all everyday orexotic spontaneous physical or chemical happenings, some type of energy flowsfrom being localized or concentrated to becoming more spread out ordispersed. Generally, the "some type" is kinetic energy. In chemistry "motionalenergy" of molecules is preferable to the phrase kinetic energy because at any

instant when molecules are ceaselessly colliding, many may be motionless for aninstant due to head-on collision of equally energetic molecules. Motional energybetter circumscribes the entire process of energetic movement. (Energy that issupplied to a substance in phase change becomespotential energy that is partof the total energy of a system, unaltered by volume change or by temperaturechange, except at phase change temperatures.)

Potential energy in macro objects (like a rock held up in the air, or waterbehind a dam) is always hindered, i.e., kept from dispersing, until it is changedto kinetic energy. The potential energy in chemistry that we have already talkedabout is that involved in phase change; coming from or going to the

surroundings, it causes breakage or formation of intermolecular bonds and thuscan free or restrict molecular motion.. Another major type of potential energy isthe energy in chemical bonds that holds molecules together. When chemicalreaction occurs to form products whose bonds are stronger than the reactants(as in the case of hydrogen and oxygen reacting to form water), some of thepotential energy that was in the hydrogen + oxygen can spread out (in the formof molecular motional energy).

To most beginning chemistry students, this generality of energy spreadingout in chemical reactions or from hot or high-pressure systems becomes


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obvious once examples have organized the concept for them.. Most haveseen high pressure air whoosh from a punctured tire. All have seen that occurfrom a balloon. All know that hot things cool down and that hydrogen andoxygen react violently. Thus, it is not a great step to a formal description ofthose phenomena as "the dispersal of energy to a larger three-dimensionalspace" and extension to energy spreading out more readily among moreparticles rather than fewer. Then later, to an extent and depth that you choose,you can lead them to see how, fundamentally, this spatial dispersion of energyalways is due to energy being dispersed because, at any instant, it can be in oneof a much larger number of microstates (the total molecular motional energy ofa system in quantized states) than the energy was before the process orreaction occured.

In exothermic reactions some of the bond energy that is in the reactantsis transferred to products that have lesser bond enthalpy, and the remainder isdetected as thermal energy, i.e., greater molecular motion in the molecules ofthe product. This motional energy ("heat") can then transfer energy from thesystem to the surroundings. Conversely, endothermic reactions are caused by

spontaneous spreading out of some energy from the more concentrated-energysurroundings (hotter) to the lesser concentrated-energy (cooler) substances inthe system.

But what does all that "energy spreading out" have to do with entropy?

Entropy change is a measure of the molecular motional energy (plus any phase change

energy) that has been dispersed in a system at a specific temperature.

Always, motional energy flows in the direction of hotter to cooler

becausethat direction of dispersal results in a greater amount of spreading outof energy than the reverse. Entropy change, q(rev)/T, quantitatively measuresenergy dispersing/spreading out. Its profound importance is that it alwaysincreases in a "hotter to cooler" process so long as you consider the universe ofobjects, systems, and surroundings.

Exactly how entropy measures energy dispersal is mathematically simplein phase change. (It's just the H of the process/T.) Further, from Tables of

standard state entropy, the Sovalues for substances at 298 K, we can get ageneral idea of how the amount of energy that has been dispersed in substances

differs in types of elements and compounds. Finally, we can measure exactlyhow much entropy increases when substances are heated, i.e., when energy isdispersed from the surroundings to them. All other areas treated in generalchem courses are easily describable to beginning students and I will do so but the details of their calculations can be left to your text.

Entropy (change) shown in standard state tables

A standard state entropy of S0for an element or a compound is the actual


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change in entropy when a substance has been heated from 0 K to 298 K.

[However, determining S0at low temperatures is not a simple calculation orexperiment. It is found either from the sum of actual measurements of q(rev)/Tat many increments of temperature, or from calculations based on

spectroscopy.] The final value of S0in joules/K is related to (but not an exactfigure for) how much energy has been dispersed to the material by heating itfrom absolute zero where perfect crystals have an entropy of 0 to 298 K.

Thus, in our view of entropy as a measure of the amount of energy

dispersed/T, S0is a useful rough relative number or index to comparesubstances in terms of the amount of energy that has been dispersed in them

from 0 K. (For ice and liquid water, the S0at both 273 K and 298 K is listed in

Tables.) Let's consider ice at its S0273of 41 J/K. Remember, that 41 joules is

only an indicator or index of the total thermal energy that was dispersed in themole of ice as it was warmed from 0 K. (Actually, several thousand joules ofenergy were added as q(rev) in many small reversible steps.)

Is that complicated? Abstract and hard to understand? An entropy valueof a substance is very approximately related to how much energy that had to bedispersed in it so that it can exist and be stable at a given temperature!

From standard state tables we can see that liquids need more energy thansolids of the same substance at the same temperature. (Of course! Liquids at298 K have required additional enthalpy of fusion, i.e., phase change energy, tobreak intermolecular attractions or bonds present in solids so that theirmolecules could more freely move in the liquid phase. And substances that aregases at 298 K similarly had to have the enthalpy of vaporization supplied tothem at some temperature between 0 and 298 K so their intermolecular

attractions in the liquid could be broken to allow their molecules to move androtate as freely as they do in gases.) Heavy elements that are solids at 298 K(and in the same column of the periodic table as lighter elements) need moreenergy to vibrate rapidly back and forth in one place in their solid state thanlighter elements. More complex molecules need more energy for their morecomplicated motions than do simpler molecules, as do similar ionic solids: thosethat are doubly charged need more energy than do singly charged to exist at298 K.

The causes of all entropy relationships are not obvious from looking at

tables of standard entropy values, but many make a lot more sense on the basisof S0being related to the motional energy dispersed in them (plus any phasechange energy) that is necessary for a substance's existence at T. (Organicmolecules are especially good examples, but are beyond what I have time to talkabout here. See "Disorder in Rubber Banks?")

The entropy change of a substance in a phase change

We all know the Clausius definition of entropy change as dS dqrev/T


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and S = q/T in a reversible process. Let's examine such a process asexemplified by the fusion of ice to form liquid water at 273 K. (Vaporization isparallel, of course.) A very large quantity of energy from warmer surroundings,the H of fusion (6 kJ), must be dispersed within the cooler solid, but the solid'stemperature is unchanged until the last crystal of ice melts. Doesnt somethingseem wrong here? All that heat input and no increase in temperature?. Andthe reverse behavior of liquid water might seem odd to a young student: Whenwater is placed in cooler surroundings than 273 K, the same large amount of 6kJ of energy is transferred to the surroundings before the water all becomes ice.

This can be rationalized from a strictly macro viewpoint by seeing theprocess of fusion as a change of motional kinetic energy in the surroundings topotential energy in the water that has nothing to do with temperature change inwater. Then, the reverse that occurs when liquid water is placed in surroundingsthat are 272.9 K , can be understood as merely changing the potential energy ofthe water system to kinetic energy in the surroundings. (A weak analogy wouldbe the kinetic energy of a pendulum swinging up toward changing totally intopotential energy at the end of its arc, and then that potential energy changing

back to kinetic energy as the pendulum swings to its low point.)

Of course, the description of the process in molecular thermodynamics ismore detailed and far more enlightening , but our goal here is primarily to seethe macro view of thermodynamic changes.

As we said at the start, because fusion is an equilibrium process andtherefore reversible, Sice ->water= qrev/T. That qrev is simply the 6kJ of the

enthalpy of fusion of ice and thus, Sice ->water= 6000 J/273 K, or 22 J/K. The

entropychange for ice to become water, results from the amount of energy,qrev that has been spread out in the ice so that it could change it to water

divided by T. Is that mysterious? Hard to comprehend?

Admittedly, what gives entropy its great power of predicting the directionof energy flow why spontaneous energy dispersal always occurs only from ahotter to a cooler system is hidden in the apparently simple process ofdividing by T. That tremendously important and relatively invisible predictiveproperty can easily be proved to students as I'll show in a minute.

The entropy change of a substance when it is heated

[The standard procedure for determining the entropy change when a substanceis heated involves calculus that may be beyond the background of many APstudents. Qualitatively, of course, the process could be described as measuringthe amount of energy dispersed from the hot surroundings to the coolersubstance, divided by the temperature, at a very large number of smalltemperature intervals from T1 to T2 and adding all of those entropy results.]

Qualitatively, from a macro viewpoint, it is obvious that entropy mustincrease in any system that is heated because entropy measures the increase(or decrease) of energy that is dispersed to a system!


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Quantitatively, determining the entropy change of a substance (such asthe iron in a frying pan) as it is heated isn't as easy as in phase change. To keepthe process of heating at least theoretically reversible, the substance should beheated in many small increments of energy (dqrev). That way the temperature

remains approximately unchanged in each increment and the process almostreversible. This is achieved in calculus. Using the usual symbols, and integrated

over the temperature range: ST1-> T2= dqrev/T. ( 1 ) But, how do we find

out the value of dqrev?

Fortunately, the heat capacity of a substance, Cp, is really an "entropy per

degree" because it is the energy that must be dispersed in the substance perone degree Kelvin, i.e., qrev/"1 K"! Therefore, if we just multiply Cpby "the

number of degrees" (in more sophisticated terms: the temperature increment,

dT), that will give us dqrev i.e., dqrev= Cp dT.

Substituting this result in ( 1 )above gives ST1-> T2 = Cp dT/T = Cpln

T2/T1..

Then, CpT is the energy dispersed within a system when it is heated

from T1to T2.

The decrease in entropy when a system is cooled

Energy always flows from hotter to cooler

In any spontaneous process, entropy increases

If anything cools, it clearly has dispersed some of its energy to its coolersurroundings. Lesser energy in it means that its entropy decreases. The coolingof a frying pan is an example that will most easily demonstrate to students howimportant is entropy. With merely that (overused!) example of a hot iron fryingpan we can show them why "heat cannot spontaneously pass from a colder to awarmer body" Clausius' original statement for one version of the second law ofthermodynamics. That leads directly to why the universe is always increasing inentropy, another version of the second law.

Recapping our basic understanding of energy and entropy: Energyspontaneously disperses from being localized to becoming spread out, if it is nothindered from doing so. Entropy change measures that process how widelyspread out energy becomes in a system or in the surroundings by therelationship, q(rev)/T. (Let's symbolize a high temperature by a bold T, and alower temperature by an ordinary T.)

If the pan (system) is hotter than the cool room (surroundings), and q(an amount of motional energy, "heat") m i g h t flow from the pan to the room orvice versa, the entropy changes would be: pan, q/Tsysand q/Tsurroundings.

Then, when q is divided by a large number, i.e., by the bold Tof the pan, theresult is a smallerentropy change than if q is divided by a small number, i.e., the


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ordinary T of the surroundings, q/T, so there is a larger entropy change in thesurroundings.) (For simplicity in quickly first bringing this conceptual point to aclass, perhaps avoid numbers. If you feel that numbers are better, at least avoiddimensions by identification of q = 1, T = 100, and T = 1 so that 1/100 in thehot pan is obvious smaller than 1/1 in the cool surroundings!)

Now, a larger entropy change wherever it occus means that energywould be more widely spread out there. Thus, because energy spontaneously

becomes more spread out, if it is not hindered, the q will move from the hotterto the cooler (from a smaller entropy state to a larger entropy state , from ourhot pan to our cool room.) This is universally true, as stated by Clausius, asis our common human experience, and as quantified by the relative entropychanges in hotter and cooler parts of this "frying pan - cool room universe". (Itshould be emphasized that it is true even under conditions in which the processof transferring energy is essentially reversible, i.e., when the difference betweenq/Tsysand q/Tsurroundings is very small.)

Finally,the spontaneous increase in entropy in the cooler room

part of this universe is greater than the entropy decrease in the hotpan part of this room-pan universe. The net result is an increase inentropy in the whole universe, the predicted result for any spontaneousprocess.

Entropy as "unavailable energy"

This is a note to clarify an often quoted but confusing sentence aboutentropy, "Entropy is unavailable energy". The sentence is ambiguous, eitheruntrue or true depending on exactly what is meant by the words. As any of uswould predict, the energy q within even a faintly warm iron pan at 298 K,

measured by its entropy per mole So, will spontaneously cause a 273 K ice cubeplaced in it to begin to melt. In this sense the pan's entropy represents instantlyavailable energy and the sentence appears untrue.

However, if any amount of energy is transferred from the 298 K pan, thepan no longer has enough energy for its q/T value to equal the entropy neededfor that amount of iron to exist at 298 K. Thus, from this viewpoint, thesentence is true, but tricky: We can easily transfer energy from the pan. It'snot "unavailable" at all except that when we actually transfer the slightest

amount of energy, the pan no longer is in its original energy and entropy states!For the pan to remain in its o r i g i n a l state, the energy is unavailable..

"Entropy is unavailable energy" or "waste heat" is also ambiguous inregard to motional energy that is transferred to the surroundings as a result of achemical reaction. That energy/T is considered an entropy increase in thesurroundings (because it is energy that is spread out in the surroundings and nolonger available in the system). However, it is completely available for work inthe surroundings or transfer to anything there at a lower temperature; it just isno longer available for the process that occurred inthe system atthe original


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temperature.

The change in entropy when a gas expands

[In this section concerning gas expansion, as well as the next that includes fluidsmixing and a solute dissolving, I confess that I get ahead of my plan a bit. My

intent was to restrict this first presentation of entropy change as involvingenergy dispersal to a macro view. However, gas expansion so cries out for theobvious statement about molecules (that Clausius could not make in his time prior to the knowledge that molecules really existed): Here the energetic fastmoving molecules have more space in which to bounce around and spread outtheir energyrather than keeping it localized in just one flask! Certainly, anystudent who remembers the kinetic molecular theory of 5-10 chapters backwould quickly agree to that rationalization for the expansion to a larger volume.

This is a valid start for a molecular interpretation of entropy change, and perhapsall that need be told to most beginning students. However, it is really only half

the of the cause of entropy increase in any process. (This idea of two factors inany entropy change is developed for you, but in far too much detail for APstudents here.

Briefly, for your information at this point, entropy change in any process is dueto two factors: first, molecular motional energy described by the kineticmolecular theory is enabling. However, for that energy to result in entropyincrease by becoming dispersed/spread out, it must be actualized by someprocess that makes accessible additional microstates (vide infra or here) Thoseprocesses that we are now going to consider include gas expansion and thus,obvious volume change. Then, we will look at mixing with other molecules (that

amounts to separation of like molecules from each other and in that way avolume change for them). This second factor is measured by probability, andfrom the way it is counted in statistical mechanics is often associated withpositional or configurational entropy. It is an unfortunate name becausestudents get the idea that there are two kinds of entropy change when,fundamentally, there is only one a change in the number of microstates.)

When an ideal gas expands from a glass bulb through a connectingstopcock into an evacuated bulb, there is no temperature change. The initialmotional energy of the molecules does not change BUT that motional energy is

now more widely spread over a larger volume than it was originally. Accordingto our basic concept of the spontaneity of energy dispersing as widely as it can,if it is not hindered, no further explanation seems to be needed. The stopcockhindered the spontaneous expansion of the gas but when it was opened, the gasexpanded into the evacuated bulb that's just one more example of energydispersing. Of course, we can also see that the entropy of the gas must haveincreased because of our corollary concept, namely, when the motional energyof a substance becomes more dispersed, its entropy is greater. From a macroviewpoint, nothing could be simpler than deducing that entropy increases whenan ideal gas expands.


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However, quantitatively, the process of determining exactly how much theentropy has increased seems to run into a brick wall: the energy that wenormally associate with motional energy has not changed in the expansion, i.e.,q has to be 0. So how can we measure entropy by q(rev)/T. The problem isthat the sudden expansion of the gas was irreversible, not reversible at eachmoment (as was phase change, and heating or cooling a substance). If we wantto measure entropy change by using S = q(rev)/T, we must find somereversible process involving dq

rev.

The solution to that problem is to reversibly compress the gas back to itsoriginal volume. The energy required to do this will be equivalent in magnitude,just opposite in sign, to the energy dispersed in the spontaneous expansion intothe vacuum. As shown in many texts, the work, w, done to compress a gasreversibly is - w = n RT ln V2/V1 and since q = - w, S = q/T = nR ln V2/V1 .

The change in entropy when gases mix and liquids mix

When a bulb containing one gas is connected to one with a different gasand the stopcock between them is opened, the two will begin to mix slowly andwill continue until each gas has thoroughly mixed with the other. Each gas hasincreased its volume with no change in temperature (if they are ideal gases thatdo not interact) and no change in the initial motional energy(or phase changeenergy)of each. The situation is exactly like a gas expanding into a vacuum.The volume of each has increased. The initial motional energy of each gas hasbecome more spread out in that larger volume and so each has increased inentropy.

The mixing of two ideal liquids can be viewed similarly but it isquantitatively different because the volume resulting from two liquids mixing isusually not exactly the sum of the two initial volumes. Nevertheless, the conceptof spontaneous mixing because of energy spreading out still applies. The initialmotional energy of each liquid has been dispersed in the final larger combinedvolume and so the entropy of each has increased. ( An old idea that there is aspecial "entropy of mixing" is an error.)

Molecular thermodynamics. The Boltzmann entropy equation.

The molecular dispersal of energy, molecular thermodynamics, isquantitatively treated by Boltzmann's relation of entropy to microstates and bythe quantization of molecular energy in quantum mechanics. But we are talkinghere today about the simple.direct. presentation of entropy in an AP orfirst-year college/university class. It must be the instructor's choice for his orher own class of beginners as to how far to go into many details of Boltzmann'sdevelopment of entropy, S = kBln WFinal/WInitial. (It will be intensively

presented in the physical chemistry course.) I will describe a few specifics in thissection, primarily as background information for instructors. Some generalitiesabout a simple view of microstates are in the "sample lecture to all chemistry


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classes" in the next section. These can perhaps be used in most AP or collegechemistry class without exceeding the students' abilities.

A sidenote: One current general chemistry text takes about 4 pages tointroduce entropy via Boltzmann. That is absurd just one moreunnecessary burden on an already overloaded beginner. The entireapproach to entropy up to this present point plus some simple generalities

about molecular thermodynamics could be presented far moreunderstandably in 4 text pages.

Saying that (and underscoring that most of the introduction to entropyprior to this section on molecular thermodynamics is usable in any class in whichentropy has been mentioned as "disorder" in the past), I think that you shouldbe aware of some conclusions from quantum mechanics related to entropy andthe significance of the Boltzmann equation. This is the content of the followingindented paragraphs. (Additional backgound is now athttp://entropysite.oxy.edu/microstate/index.html ) You can best judge whether

you should share them as "enrichment" with all of your students, with only aselect few, or with none. Molecular thermodynamics is essential in a moderndescription of entropy. It is the fundamental basis for my describingspontaneous energy dispersal as the key to understanding entropy. (But thatword dispersal has a more precise meaning in molecular thermodynamics thanjust spreading all over three-dimensional space as we have been using it inmacro thermodynamics.)

Electromagnetic energy is quantized in the same sense thatyou taught your students about photons being the quantized units oflight energy. Similarly, all types of energy are quantized, including the

energy associated with the various modes of molecules' motions. In asection to follow that discusses the expansion of a gas into a vacuumand fluids mixing from a molecular and quantum mechanical viewpoint,the explanation is properly focused only on the energy levels accessibleto the energies of the molecules. This is adequate and correct. Therationalization I suggested for medium or lower level students, namelythat vigorously moving energetic molecules would be expected to moveinto a larger volume of three-dimensional space if they were not hindredfrom doing so is a useful picture, but it is just a start. It is superficialbecause a molecular thermodynamic view of entropy change shows thatentropy change consists of two essential factors: motional molecularenergy as enablingand any process that changes the number ofmicrostates as actualizing. (If that process leads to an increased numberof microstates, it is spontaneous because it results in increasedentropy.)

The motional energy with which we are concerned in discussingentropy consists of the combined energy of translation, rotation, andvibration. (Phase change energy is motional energy of thesurroundings that has been supplied to or released by the system due toits potential energy of intermolecular bonding. It does not change in any


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of the entropy-involving processes we discuss other than phase changeitself.) Motional energy is present in a substance because of the transferof heat from the surroundings as the substance is warmed above 0 K.)The difference between rotational and between vibrational energy levelsof molecules can be seen in the molecular spectra of rotation andvibration. That between translational levels is too small to detectordinarily so it is often considered to be continuous, the equivalent ofmore than septillions of individual levels. Therefore, because threeindependent energetic variables are involved, each molecule of anysubstance at any temperature above 0 K can be in one of an extremelylarge number of different energy levels. (Of course, input of energy thatresults in a higher temperature greater molecular motion makesadditional energy levels become accessible for a molecule's energy.Similarly, in phase changes such as melting and vaporization wheremolecules can have increased opportunities for motion or when a soluteis dissolved in a solvent, large numbers of additional energy levelsbecome accessible.)

A microstate can be defined as one arrangement of all theenergies of all the particles (each on one of their many possible energylevels at one instant) that together have the total motional energy plusthe phase change energy of a whole system. Then considering howmany molecules there are in a mole, we can sense that there are a trulyunimaginable (though numerically expressible) number of microstates inthe usual chemical system at ordinary temperatures. (The quantity is of

the order of 101,000,000,000,000,000,000,000,000 . To give you a sense of

its magnitude: There are probably less than 10100atoms in the entireuniverse.) Here again is a link to extended descriptions of microstates.

A system has its total energy the energy of each of its manymolecules arranged in some particular distribution on a giganticnumber of energy levels at one instant. This would be one microstate. Inthe next instant the system is in a different microstate. This is becauseeven a single collision of two molecules usually changes their energies,and therefore this one change makes the total arrangement of all themolecular energies different than it was an instant before a differentmicrostate. (Considering the number of molecules in a mole, you canhave a slight appreciation of how many different microstates might bepossible without any change in the total energy of a system of a mole of

molecules!) An increase in the number of accessible microstatesfor a system of molecules results in an entropy increase because thena system's energy can be more dispersed or spread out in this veryprecise sense: if there are more accessible microstates for a system,t h e r e a r e m o r e c h o ic es o f d i f f e r e n t m i cr o s t a t e s i n w h i ch t h e s y st e m m ig h t b e a t t h e n e x t i n s t a n t . That is energy dispersal, agreater number of possibilities of arrangements of the energy of thesystem. That is, of course, the opposite of energy localization (ofhaving fewer and fewer choices of arrangements of the energies in thenext instant with the ultimate being only one arrangement, the


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situation at absolute zero.) Energy dispersal in terms of microstatesdoes NOT mean that the energy in any way is' smeared' or spread overmany microstates. That is impossible, of course, because the totalenergy of a system is always present each instant in a singlearrangement, a single microstate.)

Illustrations in textbooks that purport to show microstates asmarbles in various boxes, or exemplify them as molecules in various

locations in space are specious and misleading in that microstates areshown as devoid of energy. Microstates consist of the energiesof all themolecules in a system considered to be on quantized energy levels.Even though in statistical mechanics, in order to count the numbers ofmolecules different arrangements, the molecules are placed in cells andconsidered as locations in 3-D space, it is the different arrangements oftheir energies (the numbers of microstates) that are actually beingcounted combinatorially. More on this here.

(For a detailed development of the nature of microstates, see "What is a

microstate?".)

In the equation attributed to Boltzmann, S = kBln WFinal/WInitial, the W stands for the number of microstates in a system. Using

heating of a substance as just one example of the equation's pertinence,when a substance is warmed, the increased energy dispersed in it allowseach molecule to move over a greater range of speeds in its manycollisions. Its occasionally greater energy can access higher energylevels. Thereby, the number of accessible energy levels on which fromone to a large number of the molecules energies may be at one momentone moment increases enormously. (Of course, we can think of or draw

energy levels on the board for students with dots for moleculesthemselves on these lines.) In turn this results in an enormously greaterincrease in the number of microstates because each microstate is butone arrangement of all the molecular energies whose total energy isthat of the system. All the energy now in the warmer substance has thepotential of being in any one of many more microstates than it had beenin. Energy dispersion, in the sense of there being more choices forthe system's energy to be arranged if there is an increase in the numberof microstates, is the fundamental reason for an increase inentropyin any change in a system, not only from warming, but fromphase change, from volume increase, from forming a solution. Thegreater the number of microstates for a system after someprocess, the more its entropy has increased. This is whyBoltzmann's equation is essential in molecular thermodynamics.

A p o s s ib l e a p p r o a c h t o m i c r o s t a t e s f o r a l l c h em i s t r y c l a ss e s

Now, returning from our survey of molecular thermodynamics withthe powerful support of the conclusion in boldface type above, we can


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describe it simply to students in the examples I have already discussed.In the paragraphs below, I'll try to imagine my teaching a chemistry classthat could be AP or below AP level to whom entropy in macrothermodynamics has already been presented somewhat as on earlierpages. (Although the following is informal, it needs student interruptionsto come alive and be most useful to students. You can do much better,I'm sure! I will insert a "disclaimer to students" in double brackets thatyou may disdain as well as discard.)

"The energy q that we've been talking about when we learned thatentropy change was q(rev)/T is the energy of moving atoms ormolecules. What if you heat an iron pan? That means you put the q offast moving atoms or molecules from a flame into making the iron atomsin the pan speed up their fast-jittering, vibrating, moving almost in thesame place. Gases like oxygen and nitrogen in the air in our room aremoving at an average of about a thousand miles an hour. In a cold icecube, the water molecules cant move much more than iron atoms in acold pan, but yet they are rapidly vibrating. But fast or slow, hot or cold,

the energy q of those moving molecules in any substance is quantized that is, it isn't like a continuous flow of water from a faucet. Moleculesthat are moving are energetic and that energy is in bunches or units orpackets. Remember when we learned that Einstein proposed that light isquantized? He found that light could be considered to be in packets andthose were named "photons". Then also, do you remember how we sawthat the energy of electrons in the hydrogen atom was quantized onlyon specific energy levels? The energy of moving molecules theirtranslational energy (and other kinds of movement) is like that, onspecific energy levels.

[[Now relax you don't have to remember any detailsaboutwhat I'll be saying for the next couple of minutes, but I hope you'll just ageneral feel for what "microstates" of molecules areand why how manyof them are important. At least, getting an idea about what microstatesareis essential, because from now on I'll be using that word"microstates" in explaining why entropy increases or doesn't from theviewpoint of molecules' energy. ]]

Here's an impossible "thought experiment" but try it anyway!Close your eyes and pretend you can see all the energies of the differentmolecules in a drop of water, moving every which way, at any speed from0 to 2000 miles an hour, and you can see all those energies arranged on aladder-like gazillion different energy levels! Now, quick, freeze that frameof the ultra-fast movie. That's a microstate of the water molecules at thetemperature of our room all their motions stopped with their individualenergies on a literally incredible number of levels. (The total energy of allthose molecules' energies on all those levels is the energy q in thatentropy equation of q(rev)/T.) A microstate is an exact arrangement of allthe energies of the molecules of a system at one instant.

Then let the molecules move for just an instant. Freeze everything


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again. That would be another microstate slightly different in energyof a couple of molecules so it would be a slightly different arrangementwith the same total energy. Don't keep goingI don't want you to gettired because even if you got all the people of the world thinking 'freeze-frames' like that every millionth of a billionth of a second for trillions timestrillions of years, they wouldn't have visualized a 'zillionth' of the numberof microstates in any substance at room temperature!

A zillionth? In the smallest drop of water you can see, there are at

least 1010,000,000,000,000,000,000 of microstates. (There are probably less

than 10100atoms in the entire universe!) Heat that tiny drop of water upjust a little and that makes even more many more microstates (thatmany more arrangements of molecules energies in any one of which atone instant the total energy of the tiny drop would be).

Now, heres the payoff: The arrangement of the motional energiesof all the molecules in any chemical (a system) at one instant is amicrostate. So, if there are more microstates (additional accessible

arrangements) made available to the molecules, there are more choicesfor the system to be in any one of them at any instant.

Here's a practical illustration: Even though we would be talkingpretty good science if we were in a tire shop and said, when ahigh-pressure tire blew out, "Hear that air energy spreading out all over",that's not the full story. What's really happening down at the molecularlevel is that the motional energy of the high-pressure air is spreading outfrom the tire BECAUSE it then has the chance of being in any one of awhole lot more microstates in the air of the shop and THATS becausethere are more microstates whenever a substance is given more 3Dspace.

Entropy increases whenever more energy is spread out in asubstance. For example, when some substance is heated, the energy hasmany more microstates in any one of which it might be at an instant --THAT'S fundamentally what entropy increase means.

When that hot iron pan cools down, then there aren't as manymicrostates in one of which its atoms can be at one moment so we saythat the energy can't spread out as much. Therefore, the pan's entropy

decreases. But the cooling-down happened because slower air moleculeshitting the hot pan were made to move a little faster. The extra energy inthose faster air molecules now has MORE microstates in any one of whichthe energy of the air can be at an instant , so the total energy of the aircan be called more spread out.. And what does more "spread-outenergy" in the air mean? An increase in the air's entropy, of course.

Any time entropy DECREASES in the universe like the hot pancooling down the energy from that part of the universe spreads outand increases the entropy of its nearby surroundings (and that includesany cooler thing or air near it). Always the increase in entropy is greater


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than the decrease elsewhere. You could guess that would be truebecause we said when we first started to talk about thermodynamics that"energy spontaneously spreads out, if it isn't hindered" and then we saidthat "entropy increases when energy becomes more dispersed or spreadout". Put the two together and you'll predict that entropy is alwaysincreasing ..

Now, I leave the students in your more capable hands, and return to adiscussion of entropy, but from the viewpoint of molecular thermodynamics

The expansion of an ideal gas into a vacuum

(See the objections to positional or configurational entropy in the box at theend of the next section, The spontaneous mixing of fluids)

Students who know that entropy measures how spread out is the energy

in a system readily accept the qualitative molecular conclusion about entropyincrease when a gas expands into an evacuated bulb. Energetic movingmolecules? Allowed to go into a larger volume? What else a spreading outof energy deal: entropy increases because the molecules motional energy (withthe systems phase change energy) becomes more spread out/dispersed in thelarger volume. (Of course, that should be followed up by developing whyreversible restoration of the system is an essential quantitative corroboration ofthe entropy increase, the qrev/T for the irreversible change .)

The better qualitative answer to this gas expansion question comes from

a more detailed analysis via molecular thermodynamics, especially ifone factfrom quantum mechanics is added: The energy levels of a particle in a boxbecome closer together, more dense, the larger is the box. Therefore, we canconclude that the number of accessible energy levels for molecular energieswithin any small energy span increase when the volume of a system increases.So with many more energy levels for molecules' energies, there must be agreatly increased number of newly-accessible arrangements of those energies i.e., many, many more microstates for the system in the new larger volume thanin the original space. Thus, with spontaneous gas expansion, the entropyincreases because the original energy now has many more microstates in anyone of which it might be at any one instant i.e., the system's energy is

dispersed, in terms of microstates.

The spontaneous mixing of fluids

Two different ideal gases placed in two connected bulbs will mixspontaneously when the stopcock between the two bulbs is opened. The initialpressure and temperature will be unchanged. Thus, such a spontaneousisobaric isothermal process must be due to an entropy increase. The reason isnot that there is an "entropy of mixing" for ideal gases. Rather, it is simply an


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entropy increase due to greater dispersion of the molecular energy of eachcomponent throughout the new larger volume of the two bulbs completelyparallel to the free expansion of a gas into a vacuum.

(The examples are far more interesting to a class than gas mixtures, I believe,except perhaps for perfume across a room! This is where a food dye in watercan be demonstrated, and where we have an explanation for cream mixing incoffee, especially in ideal experiments where there we can say there is no fluid

movement or convection.)

In both gas and liquid mixing, the motional energy of each component of themixture has greater volume in which their energies can be dispersed morewidely. Therefore, the entropy of each increases. However, the new volume ofthe mixture of liquids is not simply the total of the components volumes as isthe case for gases (initially at equal pressures) Instead, the entropy change iscalculated on the relative number of moles of each type of liquid in the mixture.Even though the calculation is more complex, the fact is simple and clear anyspontaneous mixing allows the molecules of each component in the mixture to

spread out its energy more widely and a components entropy increases.

Calculations of entropy change in liquid mixtures are based on statisticalmechanics wherein a model of the relative quantities of components in themixture is constructed by placing those quantities of molecules(representing real and energetic molecules!) in cells in three-dimensionalspace. Because a cell is considered located at a position or configurationin space, any calculation of the number of those cells compared to theone configuration of the unmixed component is often called positionalor configurational entropy from Boltzmanns S = kBln Ways/1. This is

unfortunate in texts that so identify the results because each cell in agiven position or having a given configuration is actually a Way, amicrostate, an arrangement of the total molecular energiesfor themixture! But if the word positional is used, entropy change of mixing isseemingly completely different from thermal entropy change eventhough both are measured by change in the number of microstates. Seehttp://entropysite.oxy.edu/#calpoly for a clarification of this error.)

The increase in entropy when a solution is formed from a solute and a pure solvent

Osmosis and other colligative effects

All colligative effects are due to the increased entropy of the solvent in asolution as compared to the pure solvent alone. Probably, to most classes, thisshould be presented simply as a finding or a fact. To classes with superiorstudents, the preceding detailed analysis of the cell model in statisticalmechanics could be shared or summarized. An entropy increase occurs in asolvent even if only a small amount of ideal solid solute (no heat effects) wereadded to form a solution. This is because those solute molecules are throughoutthe solution, affecting the nature of the interaction of solvent molecules with one


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with one another. No longer does every solvent molecule have only other solventmolecules around it. Some therefore are as separated from each other as thoughthey were in a larger volume and that means that these energetic moleculeshave their energy more dispersed than in pure solvent. Their entropy hasincreased. Thus, the entropy of a solution is increased to an extent that isdependent on the number of moles of solute that have been added. Because thesolvent molecules in a solution have a larger entropy then when in pure solvent,they less tend to "escape" from their greater entropy state in the solution to avapor phase or to a solid phase than from the pure solvent.

Then colligative effects such as osmosis are easily explained. They allinvolve an increase in entropy in the solvent molecules if they are in a solution. .If a membrane permeable only to solvent molecules is placed between somesolvent and a solution of it, the solvent will spontaneously move through themembrane to the solution side. Why? Because if its molecules go into thatsolution, the entropy of the solvent molecules will increase; their energybecomes more widely dispersed in the solution than it was in the pure solvent.Change will take place if an entropy increase can occur due to that change.

The elevation of the boiling point of a solution that contains a non-volatilesolute is also caused by an entropic effect. Because the solvent has a higherentropy in the solution than the pure solvent at its ordinary boiling point, thereare not enough solvent molecules moving from the liquid solution to the vapor atthat usual boiling point temperature to equal the atmospheric pressure of 760mm. (In some texts it is said merely that the solvent's "escaping tendency" islowered in a solution with little or no explanation. However, this omits the basiccause of the phenomenon, the greater entropy of the solvent when it is in asolution.) This lower vapor pressure (fewer molecules escaping from thesolution) at the normal boiling point can only be overcome by increasing the

temperature of the solution and thereby the average energy of the molecules init, including the solvent molecules. Then, at some temperature above the usualboiling point, enough solvent molecules will be leaving the solution so that anequilibrium at 760 mm will be established between the solution and the solventvapor.

The depression of the freezing point of a solution containing a solute thatis insoluble in the solid phase of the solvent is similarly caused by an entropiceffect. The solvent in the solution has a larger entropy value than the puresolvent. Therefore, unlike the pure solvent and solid being at an equilibrium forcrystallization to occur at the solvent's normal freezing point, the solvent in thesolution has too much entropy too little "escaping tendency" to leave thesolution and form the intermolecular bonds of crystals at that usual freezingtemperature. Accordingly, the solution must be cooled so that there is lessenergy dispersed within it, fewer accessible microstates in it and its entropythereby decreased. As the temperature of the surroundings is lowered and theentropy of the atmosphere and the solid ice decrease more slowly than does thesolution, equilibrium between the liquid and solid phases is established forcrystallization at some point below the normal freezing point of the pure solvent.Then, freezing can occur as energy continues to be dispersed from the solutionto the cooler surroundings.


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Chemical Reactions how much and "how widely energy is dispersed in the Gibbs

free energy equation

G is "free energy"? What does that mean?

"Free energy", represented by G in the Gibbs equation, G = H - TS,is said to be the maximum non-expansion work that can be obtained from a

process (in a system at constant temperature). However, calling it an "energy"has been vigorously criticized because all types of energy are conservedaccording to the First Law of Thermodynamics whereas G is not conserved.What does that mean?

From what we have been considering here, it is easy for us to see why Gis not a "usual" kind of energy that is described by the First Law, just as the qrevin the qrev/T of entropy isn't a "usual" energy. (In q rev/T, qrevis energy

defined by its being involved in a specific kind of action: It is energy that hasbeen, or could be, dispersed in an equilibrium situation and can be directlyrelated to energy dispersal in non-equilbrium processes.)

The nature of G can be shown if we divide the Gibbs equation by -T, thenit becomes

-G/T = -H/T + S. Look at that: Every term in the equation is now anentropy expression! Starting from the right of the equation, S is the entropychange of the reaction in the system due to S products- S reactants. The -H/T is

the entropy change of the surroundings due to energy dispersed from thereaction to the surroundings. And, finally, the -G/T is the entropy change ofthe universe (surroundings plus system) but note! G (or, showing that it

comes from the system, -

G) therefore is the net dispersed or dispersibleenergy due to the reaction occurring inside the system.

That's why G isn't conserved and isn't a "usual" energy. G is energythat is being measured by how much it spreads out/T or by how spread out/Tit becomes. Unlike energy in general, but like all energy whose entropy effectswe have talked about, we now can see it as "entropy energy" energy that isbeing or can be dispersed at a specific of temperature, T, as a result of achemical reaction.

Just changing terminology, to "dispersible energy" rather than "free

energy", is not the point. The important idea is that if we focus on energy flow,we can have and can help our students to have a better sense of what entropymeans in this fundamental equation. Entropy isn't a complex word or notunderstandable, but rather a universal working tool.

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Last revised and updated: November 2005

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